Problem
a. Write down the probabilistic model for the Gaussian SLAM problem with K landmarks.
b. Derive the equations for a collapsed bootstrap-sampling particle-filtering algorithm in Fast SLAM. Show how the samples are generated, how the importance weights are computed, and how the posterior is maintained.
c. Derive the equations for the posterior collapsed-particle-filtering algorithm, where x(t+1)[m] is generated from P(X(t+1) | x(t) [m], y(t+1)). Show how the samples are generated, how the importance weights are computed, and how the posterior is maintained.
d. Now, consider a different approach of applied collapsed-particle filtering to this problem. Here, we select the landmark positions L = {L1, . . . , Lk} as our set of sampled variables, and for each particle l[m], we maintain a distribution over the robot's state at time t. Without describing the details of this algorithm, explain qualitatively what will happen to the particles and their weights eventually (as T grows). Are there conditions under which this algorithm will converge to the correct map?